JP2019512134A - Method and system for solving Lagrange duals of binary polynomially constrained multinomial programming problems using a binary optimizer - Google Patents

Abstract

Disclosed are methods and systems for solving Lagrange duals of binary polynomial constrained problems (BPCPPP). The method comprises the steps of obtaining BPCPPP; providing a set of Lagrange multipliers iteratively until convergence is detected, and an unconstrained binary quadratic programming problem representing the Lagrange relaxation of BPCPPP in these Lagrange multipliers ( Provide UBQPP, unconstrained binary quadratic programming problem), provide UBQPP to the binary optimizer, get at least one corresponding solution from the binary optimizer, and set a new Lagrange multiplier using at least one corresponding solution And after convergence, providing all corresponding known best principal duals and known best feasible solutions. [Selected figure] Figure 1

Description

  The present invention relates generally to computing. More precisely, the invention relates to a method and system for solving a Lagrange dual problem corresponding to a binary polynomially constrained polynomial programming problem using a binary optimizer.

This application is based on Canadian Patent Application No. 2,921,711 filed Feb. 23, 2016 and US Patent Application No. 15 / 051,271 filed Feb. 23, 2016 Claim the priority of

  This application is related to Canadian Patent No. 2,881,033.

  Background information on Lagrangian duals and duality of the quadratic binary programming problem can be found in US Pat.

  In duality theory, several different types of dualization and duality problems have been proposed. One type of duality problem is the Lagrange duality problem. A detailed description of Lagrange duality theory is disclosed in [1]. For the application of the Lagrange approach in discrete optimization, see Non-Patent Document 2 and Non-Patent Document 3.

  Several methods have been proposed as methods for solving the Lagrange dual problem, for example, as described in Chapter 3 of Non-Patent Document 1, the undergradient method, the outer Lagrange linearization method, and the bundle method. Etc. The difficulties in obtaining efficient implementations in these algorithms arise in connection with the desire to find an efficient method for solving difficult non-linear integer programming problems at various stages of these methods. For example, solving the single-constraint second-order 0-1 knapsack problem using an efficient branch-and-bound method based on Lagrange duality, as described in 111.5 of Non-Patent Document 1 Can. However, the proposed method can not be generalized to multidimensional knapsack problems.

  Patent Document 1 discloses a method for solving Lagrange duals of a constrained quadratic programming problem using a quantum anila. The disclosed method is based on the outer Lagrange linearization method. Unfortunately, the limitations of the method when used with a quantum anila include the fact that error-prone unconstrained quadratic optimization problems can be manifested during processing.

  What is needed is a method and system for solving the Lagrangian dual optimization problem that overcomes at least one of the above identified shortcomings.

  The features of the present invention will become apparent by reference to the disclosure of the present specification, the drawings and the following description of the present invention.

Canadian Patent No. 2,881,033 US Patent Application No. 2006/0225165 U.S. Patent Application No. 2008/0218519 U.S. Patent No. 8,655,828

“Nonlinear integer programming”, Duan Li and Xiaoling Sun “A survey of Lagrangean techniques for discrete optimization”, Jeremy F. Shapiro, Operations Research Center, Massachusetts Institute of Technology, Cambridge, Massachusetts “Lagrangean relaxation for integer programming”, A. M. Geoffrion, Mathematical Programming Study 2 (1974) 82 114, North-Holland Publishing Company Farhi, E. et al., “Quantum Adiabatic Evolution Algorithms versus Simulated Annealing”, arXiv.org: quant ph / 0201031 (2002) McGeoch, Catherine C. and Cong Wang, (2013), “Experimental Evaluation of an Adiabatic Quantum System for Combinatorial Optimization” Computing Frontiers, May 14, 16th, 2013 (http://www.cs.amherst.edu /ccm/cf14-mcgeoch.pdf) “A practical heuristic for finding graph minors”, Jun Cai, William G. Macready, Aidan Roy

  According to a broad aspect, the following method is disclosed: a method for solving Lagrange duals of binary polynomial constrained problems (BPCPPP) The method provides a set of Lagrange multipliers, iteratively until convergence is detected, with the step of obtaining a binary polynomially constrained polynomial programming problem using a processing unit An unconstrained binary quadratic programming problem (UBQPP, unconstrained binary quadratic programming) that provides an unconstrained binary quadratic programming problem representing Lagrange relaxation of a binary multinomial constrained multinomial programming problem in a set of Lagrange multipliers problem) to the binary optimizer, obtain at least one corresponding solution from the binary optimizer, and at least one corresponding Generating a new set of Lagrange multipliers using; and, after convergence, the known best principal dual and binary polynomial constraints of the Lagrange dual of the corresponding binary polynomially constrained polynomial programming problem Providing all of the known best feasible solutions of the multinomial planning problem.

  According to a broad aspect, the following method is disclosed: a method for solving Lagrange duals of a binary polynomially constrained polynomial programming problem, wherein the binary polynomial constrained polynomials Step of obtaining the planning problem and iteratively until convergence is detected: provide a set of Lagrange multipliers and represent the Lagrange relaxation of a binary multinomial constrained multinomial programming problem in these Lagrange multipliers. Provide a binary quadratic programming problem of, provide an unconstrained quadratic programming problem to the binary optimizer, obtain at least one corresponding solution from the binary optimizer, and use at least one corresponding solution Generating a new set of Lagrange multipliers and knowing Lagrange duals of the corresponding binary polynomially constrained multinomial programming problem after convergence is detected Providing all of the known best feasible solutions of the best primal dual and binary multinomial constrained multinomial programming problems being done.

  According to an embodiment, the acquisition of a binary polynomially constrained multinomial programming problem involves acquiring data representing an objective function that is a polynomial, acquiring data representing a polynomial equality constraint, and a polynomial inequality constraint. Obtaining data representing H.

  According to an embodiment, the binary multinomial constrained multinomial planning problem is obtained from at least one of a user, a computer, a software package and an intelligent agent.

  According to an embodiment, acquiring the binary polynomially constrained polynomial programming problem further comprises initializing software parameters and acquiring a step size subroutine.

  According to an embodiment, initializing the software parameters may be a general degree reduced form (GDRF, generic degree reduced form) of the general Lagrange relaxation of the binary polynomially constrained polynomial programming problem, the original variables. And providing as a parameterized family of binary quadratic functions parameterized with Lagrange multipliers using auxiliary variables.

  According to an embodiment, initializing the software parameters provides a general embedding for the general order reduction form of the general Lagrange relaxation of the binary polynomially constrained polynomial programming problem, and the solution Providing an embedded solver function to provide a list of 、, providing one of an initial value and a default value for the Lagrange multiplier, providing an error tolerance for the convergence condition, the total number of iterations Providing an integer representing the limit value for and the limit value for non-improved iterations.

  According to an embodiment, an initial Lagrange relaxation of the binary polynomially constrained polynomial programming problem is generated using an initial Lagrange multiplier.

  According to an embodiment, the initial degree reduction form of Lagrange relaxation of the binary polynomially constrained polynomial programming problem is solved using a quantum aniler, and at least one corresponding solution is obtained.

  According to an embodiment, at least one corresponding solution is used to generate a subgradient of Lagrange duals of a binary polynomially constrained polynomial programming problem.

  According to an embodiment, at least one corresponding solution is used to generate at least one outer approximation cut for linear reformulation on Lagrange duals of a binary polynomially constrained polynomial programming problem.

  According to an embodiment, providing corresponding solutions to Lagrange duals of a binary polynomially constrained polynomial programming problem involves storing the corresponding solutions in a file.

  According to a broad aspect, the following digital computer is disclosed: a digital computer: a central processing unit; a display unit; a communication port for operatively connecting the digital computer to the binary optimizer And a memory unit comprising an application for solving Lagrange duals of a binary polynomially constrained polynomial programming problem, the application comprising: instructions for obtaining a binary polynomially constrained polynomial programming problem And iteratively provide a set of Lagrange multipliers and represent the Lagrange relaxation of a binary polynomially constrained multi-term programming problem in these Lagrange multipliers, and provide an unconstrained binary quadratic programming problem, and a communication port Provide an unconstrained quadratic programming problem to the binary optimizer; using the communication port to An instruction for obtaining at least one corresponding solution from the miser and generating a new set of Lagrange multipliers until convergence is detected using the at least one corresponding solution, and corresponding after convergence is detected Provides all of the known best primal duals of Lagrange duals of binary polynomially constrained polynomial programming problems and all known best feasible solutions of binary polynomial constrained polynomial programming problems A digital computer, comprising: a memory unit including instructions for: a data bus for interconnecting the central processing unit, the display unit, the communication port, and the memory unit.

  According to a broad aspect, the following non-volatile computer readable storage medium is disclosed: non-volatile computer readable storage medium for storing computer executable instructions, said instruction executing a binary Allowing a digital computer to perform a method for solving Lagrange duals of a multinomial constrained multinomial programming problem, which comprises: obtaining a binary multinomial constrained multinomial programming problem, convergence is detected Until it is done iteratively: provide a set of Lagrange multipliers, and provide an unconstrained binary quadratic programming problem representing the Lagrange relaxation of the binary polynomially constrained polynomial programming problem in these Lagrange multipliers, Provide an unconstrained quadratic programming problem to the binary optimizer and obtain at least one corresponding solution from the binary optimizer, The step of generating a new set of Lagrange multipliers using at least one corresponding solution, and after convergence is detected, the Lagrange duals of the corresponding binary multinomial constrained multinomial programming problem are known Providing all of the known best feasible solutions of the best principal dual and binary polynomially constrained polynomial programming problems.

  According to an embodiment, the binary optimizer comprises a quantum aniler.

  An advantage of the disclosed method for solving Lagrange duals in BPCPPP is that it is less sensitive to errors in the quantum system. These errors can be caused by noisy qubits used in some implementations of quantum anilers (eg, D-Wave Systems). Thus, the disclosed method significantly improves the processing content of systems including quantum anilers and solves the technical problems caused by noisy qubits used in some implementations of quantum anilers. It will be done.

  Another advantage of the disclosed method is to provide a method of using Lagrange duality in various applications, for example, using quantum anila for integer programming problems and mixed integer programming problems, the limitations of Lagrange systems It is mentioned to search.

  Another advantage of the disclosed method is that it improves the process content of the system for solving BPCPPP Lagrange duals.

  BRIEF DESCRIPTION OF THE DRAWINGS Embodiments of the present invention are illustratively shown in conjunction with the accompanying drawings in order to facilitate the understanding of the invention.

  Further details of the invention and their advantages will become apparent from the detailed description that follows.

FIG. 6 is a flow chart illustrating an embodiment of a method for solving Lagrange duals of binary polynomial constrained programming problem (BPCPPP) using a quantum anila. FIG. 1 is a schematic diagram of a system that can implement a method for solving Lagrange duals of BPCPPP using a quantum anila, wherein the system comprises a digital computer and a quantum anila. FIG. 7 is a schematic diagram for an embodiment of a digital computer used in a system for solving Lagrange duals of BPCPPP using a quantum anila. Figure 2 is a flow chart illustrating an embodiment for providing a BPCPPP. FIG. 7 is a flow chart illustrating an embodiment of initializing software parameters used in an embodiment of a method for solving Lagrange duals of BPCPPP. FIG. 5 is a flow chart illustrating an embodiment of a proposed method as a method for updating a Lagrange multiplier.

  In the following description of the embodiments, reference to the accompanying drawings is made by way of example with reference to the examples, which illustrate the manner in which the present invention may be practiced.

Terms The terms "invention" and the like mean "one or more inventions disclosed in the present application", but this is not the case unless expressly specified otherwise.

  "Aspect", "embodiment" with indefinite article, "embodiment" without article, "embodiment" with plural article, "embodiment" with definite article, "embodiment" with plural. The terms "one or more embodiments", "some embodiments", "specific embodiments", "an embodiment", "another embodiment", etc. The above means (but not all), but this is not the case when it is explicitly specified otherwise.

  When reference is made to "another embodiment" or "another aspect" when describing an embodiment, the referred embodiment may be another embodiment (e.g. the implementation described before the reference embodiment). It does not imply that it is mutually exclusive with form), but it is not this limitation when it is specified otherwise explicitly.

  “Includes”, “includes”, and these variations mean “includes, but is not limited to”, unless otherwise explicitly stated.

  The words "indefinite article a", "indefinite article an", "definite article the" and "at least one" mean "one or more", but have been explicitly stated otherwise If this is not the case.

  The word "plural" means "more than one", but this is not the case unless otherwise specified.

  The term "herein" means "in the present application and including any which is incorporated by reference", but this is not the case unless expressly specified otherwise.

  Words such as "exempli gratis" mean "for example" and do not limit the words or phrases described. For example, in the sentence "a computer sends data (eg, an instruction or data structure) on the Internet", the word "eg" may be "data" for which an "instruction" may be sent on the Internet by a computer. It is also described that the "data structure" is an example of "data" that can be sent over the Internet by a computer. However, "instruction" and "data structure" are only examples of "data", and things other than "data structure" and "data" may be "data".

  Words such as "id est" mean "i.e." and limit the words or phrases described. For example, in the sentence "a computer sends data (ie, an instruction) on the Internet", the word "ie" is "data" such that "instructions" are sent on the Internet by a computer. Explain.

Words such as “binary polynomial constrained problems (BPCPPP)” are real polynomials y in a plurality of binary variables x = (x ₁ ,..., X _n ). It means finding the minimum value of = f (x), where the following constraints are imposed (hereinafter also expressed as "subject to"): (m may be empty set) A family of equality constraints (which may be an empty set) determined by a family of expressions g _j (x) = 0 (where j = 1, ..., m), as well as (which may be an empty set) A family of inequality constraints (which may be an empty set) determined by a family of l inequalities h _j (x) ≦ 0 (where j = 1,..., l). Here, all the functions g _i and h _j are polynomials.

The "domain" of BPCPPP means the set {0, 1} ⁿ for vectors of size n with binary entries. The "executable domain" of BPCPPP means a subset of the domain F ⊆ {0, 1} ^n, which means that all of the above m equality constraints and l inequality constraints are satisfied. The binary vector v∈ {0, 1} ⁿ is constructed.

The aforementioned BPCPPP is represented as (P), and its optimum value is represented as v (P). The optimal solution x (i.e. the vector whose objective function achieves the value v (P)) is denoted as x ^* .

BPPPPP's (P) "Lagrange function" means the following function:

The above optimization value expressed as δ _p (λ, μ) is known as the lower limit of the optimum for the original BPCPPP, ie δ _p (λ, μ) ≦ v (P) .

The term "Lagrange dual" in BPCPPP is used in the context of the following optimization problem:

  The above optimization is denoted as δ (P) and is known as the lower limit of the optimal value for the original BPCPPP, ie δ (P) ≦ v (P). This value is unique and is also referred to as the "Lagrange dual limit" for the original BPCPPP.

Terms such as "optimal Lagrangian multiplier" refer to sets of points λ ^* and μ ^* that are not necessarily unique, where the value of δ (P) is achieved in relation to the optimization problem described above.

  "Solution for the Lagrange dual problem" for the original BPCPPP points to a collection of the following information received after solving the Lagrange dual problem: (1) Lagrange duality limit; (2) Optimal Lagrange as described above A (not necessarily unique) set of multipliers; (3) a (not necessarily unique) set of binary vectors for which the optimal value of the Lagrange dual problem is obtained at a given optimal Lagrange multiplier

"Known best main dual are" (best-known primal-dual pair ) and the processing being executed δ _{p (λ,} μ) is the maximum observed value of the functions serving [delta] _p such Primal-Dual Point to the bar (x, λ, μ).

  Words such as "quantum anila" refer to systems that include one or more types of hardware that can find optimal or suboptimal solutions for unconstrained binary polynomial programming problems. means. As an example, mention may be made of a digital computer in which binary polynomial constrained numerical polynomial programming is embedded as an Ising spin model and is connected to an analog computer. The analog computer can optimize the spin configuration in the Ising spin model using quantum annealing, for example, as disclosed in the following document: Non-Patent Document 4, pp. 1-16. Examples of such analog computers are disclosed in US Pat. It should be noted that the "quantum anila" may also comprise "classical components" such as classical computers. Thus, the "quantum anila" can be either fully analog or analog-typical hybrid. It should also be noted that "quantum anila" is an embodiment of a binary optimizer.

Words such as "embed" about the binary optimization problem are assigned emb: {x ₁ , ...,
_{x n, y 1, ···,} y t} → 2 {q1, ··, points to ^qn},該割those each and auxiliary variables y of binary variables x _i a subset of the total qubit quantum annealer Passing to each of _j , connectivity between variables is to be adhered to by the connectivity of the images under their emb. For example, if the term x _r x _s appears in the GPRF of the general Lagrange relaxation of BPCPPP, then the subset of qubits corresponding to the two variables x _r and x _s is between them in the quantum aniler It should have a bond. For details of the role of embedding as described above in solving the unconstrained binary polynomial programming problem and in proposing an efficient algorithm for it, see the following document: Patent document 3 and patent document 4.

  The terms "embedded solver" etc. refer to functions, procedures and algorithms that include instructions such as: receiving unconstrained binary quadratic programming problems (UBQPP), and Performing a query on the quantum aniler using embedded embedding and returning at least one result, each result including a vector of binary entries representing the following: the provided constraint Items related to binary points in the domain of unconstrained binary quadratic programming (UBQP), items related to the value of the objective function of UBQP at that point, and items related to the number of occurrences of results in the entire number of reads

  The term "subroutine" or the like refers to a user-defined type of function, procedure or algorithm that is called iteratively by the software at any time during execution. In the presently disclosed system, the step size subroutine determines the next step size in the iterations shown in FIG. In the presently disclosed system, the embedded solver subroutine queries queries to the quantum aniler, and there are also aspects that embedding has provided.

Words such as "unconstrained binary quadratic programming problem" (UBQPP) mean searching for the minimum value of the objective function y = x ^t Qx + b, where Q is It is a symmetric and square execution sequence of size n, b is real and is also called objective function bias. The domain of the function spans all the vectors satisfying the following condition with binary entries: xεB ⁿ = {0,1} ⁿ .

  Neither the title nor the abstract of the invention should be used to limit the scope of the disclosed invention in any respect. The title of the invention of the present application and the item names in the specification are for convenience only, and should not be used to limit the disclosure content from any viewpoint.

  Various embodiments are described herein and are presented for illustrative purposes only. It is to be noted that the described embodiments are in no way limiting and are not intended as such. As is apparent from the disclosure, the disclosed invention is applicable to various embodiments. Those skilled in the art will realize that the disclosed invention can be implemented with various changes and modifications such as structural changes and logical changes. Although specific features of the disclosed invention may be described in relation to one or more specific embodiments and / or drawings, such features may be referred to in the description thereof. It should be noted that the invention is not limited to the examples in the embodiments and the drawings. However, this is not the case when it is explicitly stated otherwise.

  The invention can be implemented in various ways, including methods, systems and computer readable media (e.g. computer readable storage media). It should be noted that in this specification, these embodiments and any other form that the present invention can take may be referred to as a system or method. Components such as processors and memories that are described as being configured to perform a task may include both: a general purpose primarily configured to perform a task at a given time Components, and specific components manufactured to perform a task.

  Taking all the above into consideration, the invention will be described in the following regarding a method and system for solving Lagrange duals of BPCPPP.

  Although a quantum aniler is used in the embodiments described below, those skilled in the art will be aware that a quantum anila is one implementation of a binary optimizer.

  Referring to FIG. 2, an embodiment for a system 200 is shown, in which an embodiment for a method for solving a Lagrange dual of BPCPPP can be implemented.

  System 200 comprises a digital computer 202 and a quantum aniler 204.

  Digital computer 202 receives the BPCPPP and provides at least one solution for the Lagrange dual of BPCPPP.

  It should be noted that BPCPPP can be provided according to various embodiments.

  In one embodiment, BPCPPP is provided by a user interacting with digital computer 202.

  Alternatively, the BPCPPP is provided by another computer, not shown, which is operatively connected to the digital computer 202.

  Alternatively, BPCPPP is provided by an independent software package.

  Alternatively, BPCPPP is provided by an intelligent agent.

  Similarly, it should be noted that at least one solution for Lagrange duals of BPCPPP can be provided in accordance with various embodiments.

  According to one embodiment, at least one solution for Lagrange duals of BPCPPP is provided to a user interacting with digital computer 202.

  Alternatively, at least one solution for the Lagrange dual of BPCPPP is provided to another computer operatively connected to digital computer 202.

  In fact, one skilled in the art will be aware that digital computer 202 can be any type of computer.

  In one embodiment, digital computer 202 is selected from the group consisting of: desktop computer, laptop computer, tablet PC, server, smartphone, etc.

  Turning to FIG. 3, an embodiment for digital computer 202 is shown therein. Digital computer 202 may also be referred to as a processor in a broad sense.

  In this embodiment, the digital computer 202 comprises: a central processing unit (CPU) 302, also called a microprocessor or processing unit, a display 304, an input device 306, a communication port 308, a data bus 310, and a memory unit 312. .

  CPU 302 is used to process computer instructions. It will be apparent to those skilled in the art that various embodiments of CPU 302 can be provided.

  In one embodiment, CPU 302 is a Core i7 3820 running at 3.6 GHz (Intel (R)).

  Display 304 is used to display data to the user. It will be apparent to those skilled in the art that various types of embodiments of display 304 can be provided.

  In one embodiment, display 304 is a standard liquid crystal display (LCD) monitor.

  Communication port 308 is used to share data with digital computer 202.

  Communication port 308 may include, for example, a Universal Serial Bus (USB) port for connecting a keyboard and mouse to digital computer 202.

  Communication port 308 may further include a data network communication port, such as an IEEE 802.3 port, to allow digital computer 202 to connect with another computer via a data network.

  It will be apparent to those skilled in the art that the communication port 308 can provide various alternative embodiments.

  In one embodiment, communication port 308 may include an Ethernet port and a mouse port (eg, that of Logitech).

  Memory portion 312 is used to store computer-executable instructions.

  In one embodiment, memory portion 312 comprises operating system module 314.

  It will be apparent to those skilled in the art that various types of operating system modules 314 can be used.

  In one embodiment, operating system module 314 is Windows® 8 manufactured by Microsoft®.

  The memory unit 312 further comprises an application 316 for solving Lagrange duals of BPCPPP.

  Application 316 has instructions for obtaining BPCPPP.

  The application 316 further comprises instructions for performing the following processing: iteratively providing a set of Lagrange multipliers, providing UBQPP representing the Lagrange relaxation of BPCPPP in these Lagrange multipliers, UBQPP to the quantum anila Providing; obtaining at least one corresponding solution from the quantum aniler and generating a new Lagrange multiplier set using the at least one corresponding solution.

  The application 316 further comprises instructions to provide a solution corresponding to at least one solution for the Lagrange dual of BPCPPP if convergence is detected.

  The CPU 302, the display device 304, the input device 306, the communication port 308, and the memory unit 312 are interconnected via a data bus 310.

  Returning to FIG. 2, it should be noted that the quantum aniler 204 is operatively connected to the digital computer 202.

  It should be noted that in connecting the quantum aniler 204 to the digital computer 202, it is possible to provide various embodiments.

  In one embodiment, the quantum aniler 204 and digital computer 202 are connected via a data network.

  It should be noted that the quantum aniler 204 can be of various types.

  In one embodiment, the quantum aniler 204 is from D Wave Systems Inc. Further information relevant to this embodiment for Quantum Anila 204 is available at the following site: http://www.dwavesys.com. It will be apparent to those skilled in the art that various alternative embodiments of the quantum aniler 204 can be provided.

  More precisely, quantum aniler 204 receives the UBQPP from digital computer 202.

  The quantum aniler 204 can solve the UBQPP and can provide at least one corresponding solution. In the phase in which multiple corresponding solutions are provided, the multiple corresponding solutions may include an optimal solution and a suboptimal solution.

  At least one corresponding solution is provided by the quantum aniler 204 to the digital computer 202.

  It turns to FIG. Referring to process step 102, BPCPPP is obtained.

  Turning now to FIG. 4, an embodiment for providing a BPCPPP is shown.

As mentioned above, BPCPPP can be expressed as:

Here, the functions f (x), g _i (x) and h _j (x) are real polynomials for multiple variables.

  Process step 402 provides data representing an objective function f (x) which is a polynomial.

Process step 404 provides data representing equality and inequality constraints g _i (x) and h _j (x).

  It should be noted that for the implementation of the acquisition of BPCPPP, it is possible to bring about various embodiments.

  As mentioned above, and in one embodiment, BPCPPP is provided by a user interacting with digital computer 202.

  Alternatively, the BPCPPP may be provided by another computer operatively connected to the digital computer 202.

  Alternatively, BPCPPP is provided by an independent software package.

  Alternatively, BPCPPP is provided by an intelligent agent.

  Referring to FIG. 1, and in accordance with process step 104, software parameters are initialized.

  In one embodiment, software parameters are initialized by digital computer 202.

  Turning to FIG. 5, there is shown an embodiment or subroutine for parameter initialization and the manner in which default values are used for them.

  Process step 502 provides a GDRF for the general Lagrange relaxation of BPCPPP.

  Process step 504 provides data representing a general embedding for the general Lagrange relaxation GPRF of BPCPPP.

  In one embodiment, the embed is stored by the user in the ORACLE namespace and is ORACLE :: embedding.

  Continuing to refer to FIG. 5, and in accordance with process step 506, an embedding solver subroutine is provided.

  In one embodiment, the function is implemented by the user in the ORACLE namespace and is ORACLE :: solve_qubo.

  The input parameters of the embedded solver subroutine are pointers to instances of the ORACLE :: solve_qubo data type and represent an unconstrained binary quadratic programming problem, the corresponding bias being ignored.

  The output of the embedded solver subroutine is a pointer to an instance of the ORACLE :: result type, which represents a list of optimal solutions and suboptimal solutions for an unconstrained binary multinomial programming problem.

In the following, an example of a code snippet written in C ++ to provide a subroutine using an API (Sapi 2.0) developed by D-Wave Systems is presented. The functions and types used in this snippet are all supported by Sapi 2.0 except for two helper functions: qubo_to_sing and spin_to_binary. The former function converts UBQPP of ORACLE :: qubo type to Ising spin problem of sapi_problem type, and does this by variable conversion with s = 2x-1. The second function receives a vector array in {−1, 1} and returns a binary vector by applying the inverse transform with x = 1⁄2 (s + 1).

  In one embodiment, it can be seen that using the digital computer 202, the process of providing an unconstrained binary optimization problem to the quantum aniler 204 has been achieved.

  More precisely, in one embodiment, a token system on the Internet is used to provide remote access to the quantum aniler 204 and to provide authentication in use.

  It can be seen that in one embodiment, at least one solution is provided by the quantum aniler 204 in a table according to the instructions for using the quantum aniler 204.

  In one embodiment, the D Wave system provides at least one solution in the data type of sapi_IsingResult *, which is automatically type cast to an instance of QUBO :: result *.

  Continuing to refer to FIG. 5, and in accordance with process step 508, lower and upper limits for the Lagrange multiplier are provided.

  Note that in one embodiment, the act of serving for these double real numbers is accomplished by overriding the names ORACLE :: dual_lb and ORACLE :: dual_ub. Each of these types needs to contain a double array of size m + 1. The first m entries of these arrays represent the lower and upper bounds for the Lagrange multipliers corresponding to the m equality constraints, respectively, and the last l of these represent the Lagrange multipliers corresponding to the l inequality constraints. It represents the lower limit and the upper limit, respectively.

  Note that if these names are not overwritten, their values are initialized with default values.

  In one embodiment, the predetermined lower bound for the Lagrange multiplier corresponding to the equality constraint is −1 e 3 and the predetermined upper bound for that is +1 e 3.

  The default lower bound for Lagrange multipliers corresponding to inequality constraints is 0, and the default upper bound for that is + 1e3.

  Continuing to refer to FIG. 5, and in accordance with process step 510, an initial value for the Lagrange multiplier is provided.

  It should be noted that provision of these values as real numbers is achieved by overwriting the name ORACLE :: dual_init_val with a double array of size m + l. If this name is not overwritten, their values are initialized with default values.

  The default initial value for any Lagrange multiplier corresponding to any equality or inequality constraint is zero.

  Process step 512 provides an error tolerance for the undergraded norm of Lagrange duals. A strong duality is considered to hold if the norm of any subgradient of the Lagrange relaxation falls below this tolerance. In such cases, the Lagrange multiplier is optimal. According to one embodiment, the error tolerance is initialized to 1e-5 and stored as ORACLE :: tol, unless overridden by the user. Error tolerances are used in several places in the software.

  According to one embodiment, ORACLE :: tol is used to verify equality and inequality constraints. In particular, if the value (value) of all entries of LHS-RHS is at most ORACLE :: tol, then any system of inequality LHS ≦ RHS is considered to be satisfied. Similarly, if the absolute value (absolute value) of all entries in LHS-RHS is at most ORACLE :: tol, then any system of the LHS = RHS equation is considered to be satisfied.

  Continuing to refer to FIG. 5, and according to process step 514, a limit value for the total number of iterations is provided.

  According to one embodiment, the limit value for the number of iterations is initialized to 1e3 and stored as ORACLE :: MaxItr, unless overwritten by the user.

  If the algorithm reaches the number of iterations of ORACLE :: MaxItr at processing step 116, the algorithm is terminated and at least one known best-known prime-dual pair and at least one known The best-known feasible solution is returned.

  According to step 514, a limit is provided for the number of non-improved iterations.

  According to one embodiment, the limit value for the number of iterations is initialized to 10 and stored as ORACLE :: MaxNonImpItr, unless overwritten by the user. If the best-known Lagrangian dual value does not increase during the number of ORACLE :: MaxNonImpItr iterations, then the algorithm is terminated and at least one of the known best principal dual and At least one known best feasible solution is returned.

  According to step 516, a subroutine is provided for finding the step size.

  In one embodiment, the subroutine is implemented by the user in the ORACLE namespace and is ORACLE :: StepSize. If ORACLE :: StepSize is not overwritten by the user, then the fixed step size for the default value is used.

  In one embodiment, the ORACLE :: StepSize subroutine receives as input a double * object representing a search direction, and returns a double object representing a step size as output. .

  In one embodiment, the search direction is the undergradient of the Lagrange dual function, and the step size is a fixed value of one.

  Returning to FIG. 1, and in accordance with process step 106, UBQPP is generated which represents the Lagrange relaxation of BPCPPP (P) at the current value of the Lagrange multiplier.

  In one embodiment, it is understood that UBQPP, which represents the Lagrange relaxation of BPCPPP (P) at the current value of the Lagrange multiplier, is generated by the digital computer 202.

More precisely, in this embodiment it is understood that the generation of UBQPP is achieved by doing the following: GCPRF of the general Lagrange relaxation of BPCPPP (P) q _{λ, μ} (x , Y), replace the current value of the Lagrange multiplier with the parameters λ and μ.

  In one embodiment, UBQPP information is stored under the name ORACLE :: qubo.

  Continuing to refer to FIG. 1, and according to process step 108, the subroutine ORACLE :: solve_qubo is called as input with UBQPP ORACLE :: qubo and embedded ORACLE :: embedding as input, whereby UBQPP To provide at least one corresponding solution for.

  It is understood that in one embodiment at least one corresponding solution for UBQPP is achieved with a pointer to an instance of type ORACLE :: result.

  Continuing to refer to FIG. 1, and in accordance with process step 110, at least one solution of UBQPP is converted to a point within the BPCPPP domain (P).

  Continuing to refer to FIG. 1, and in accordance with process step 112, perform a check, thereby either of the at least one solution for UBQPP corresponds to a viable solution of BPCPPP (P). Decide.

  According to the same processing step, at least one of the known best principal dual and at least one known best feasible solution are updated.

Then, turning to processing step 114, a Lagrange relaxation undergrading is generated using at least one corresponding solution for the unconstrained binary quadratic program, ie δ _p (λ, μ).

  Turning to FIG. 6, and in accordance with process step 602, at least one corresponding binary vector within the BPCPPP domain (P) is provided.

According to processing step 604, the UBQPP is solved
The corresponding Lagrange relaxation undergradient is derived as follows:

  In one embodiment, if multiple binary solutions for unconstrained quadratic programming problems are provided, the undergradient with the smallest norm can be selected.

  In one embodiment, the derived underslope can be normalized.

  According to process step 606, the derived undergradient is provided to the step size subroutine to obtain a value for the step size.

  According to an embodiment, if the step size subroutine is initialized by the user, find the step size α by making a call to ORACLE :: StepSize.

According to process step 608, the Lagrange multipliers are updated as follows:

Here, new: new, old: old.

Returning to FIG. 1, and in accordance with process step 116, a test is performed to determine whether the best value of δ _p (λ, μ) has been improved in the previous ORACLE :: MaxNonImpItr steps. To determine

Process step 118 provides the result of the optimization if the best value of δ _p (λ, μ) has not been improved in the previous ORACLE :: MaxNonImpItr steps.

In one embodiment, the result is understood to include the set including: all known best principal duals (x ^* , λ ^* , μ ^* ) and known All the best possible solutions.

  In one embodiment, the results are stored in a file using a digital computer.

  An advantage of the method of the present disclosure is to enable an efficient method in searching for a solution of BPCPPP, which is achieved by finding the solution of its Lagrange dual using a quantum anila.

  Furthermore, it becomes clear that the method disclosed herein can improve the processing content of the system for solving the Lagrange dual of BPCPPP.

  It will be understood that the Lagrange multipliers will be updated iteratively until the end is made. If the norm of the undergrading of the Lagrange dual function is at most ORACLE :: tol, or if no improvement is seen for the best main dual after ORACLE :: MaxNonImpItr or ORACLE :: MaxItr iterations, then the program finish.

  In the following, an example of how to use the method of the present disclosure will be illustrated, and the case of application to the maximum quadratic stable set problem (MQSSP) will be described.

Let G = (V, E) define a graph for n vertices; W is a matrix representing the weighting of the edge E, a symmetric and square matrix of size n; and A Is the adjacency matrix of G. The MQSSP is formulated as follows:

By adopting a negative sign for the objective function, the maximization objective function can be described as minimization as follows:

In one example, let a graph with five vertices represent a group of five coworkers. Each colleagues pair shall be assigned the utility factor regarding cooperation between colleagues (u tility factor). For each individual, a utility factor is assigned for each outcome, and these values are on the diagonal of the matrix W. These utilities can be expressed as an upper triangular matrix:

Suppose each worker has a work shift, and for co-worker pairs with overlapping shifts, the matrix A has non-zero entries, for example:

  The problem of choosing the most productive team formation for a project, with no two team members having duplicate shifts, is an example of the MQSSP.

Given the above matrices A and W, the following BPCPPP is obtained according to process step 102:

The order reduction form of the Lagrange relaxation of this problem is the same as the Lagrange relaxation itself, since the two polynomials of the objective function and the inequality constraint in the above example are quadratic. Embedding is provided for the complete graph which is 5 in size. The following code snippet provides embedding for a complete graph of size 5 on a two-part K _4,4 -two-part chimera block, which consists of two parts, one for each The size is 4 and the first part is indexed with an integer from 0 to 3 and the second part is indexed with an integer from 4 to 7.

The lower and upper limits of the Lagrange multiplier are set as 0 and 100 respectively. 0 is assigned as the initial value of the Lagrange multiplier. The tolerance for the undergraded norm of the Lagrange dual function is set as 10 ⁻³ . Limits for the total number of iterations and the number of iterations that do not result in improvement are 100 and 5, respectively. The step size subroutine used in this example is the subroutine that results in a fixed step size of size 0.5 according to the following script:
double ORACLE :: StepSize () {return 0.5;}

When the method is started, the list for the best feasible solution and the best principal dual is initialized to an empty set. The Lagrange multipliers of the presented method are initialized at λ ¹ = 0 and the following problems are solved by the quantum anila:

The optimal solution obtained is x ¹ = ( ^{1, 1, 1, 1, 1} ), and the optimal value is −33. Since x ¹ is not feasible, the list of best feasible solutions is not updated. However, the best principal dual is updated to (x ¹ , λ ¹ ).

The undergradient of the following Lagrange relaxation is x ^t Ax = 8 for x ¹ = (1,1,1,1):

For the step size subroutine, a fixed step size of 0.5 is used. And its next Lagrange multiplier is calculated as follows:
λ ^{2 =} 0 + 0.5 * 8 = 4

The following Lagrange relaxation problems are solved by the quantum anila:

An optimal solution x ² = (1, 0, 1, 1, 0) with an optimal value of −16 is obtained. Even though the best principal dual is updated to (x ² , λ ² ), the best feasible solution is left as an empty set because x ² is not feasible.

The Lagrange relaxation undergrading in this solution is two. And its next Lagrange multiplier is calculated as follows:
λ ^{3 =} 4 + 0.5 * 2 = 5

The following Lagrange relaxation problems are solved by the quantum anila:
An optimal solution x ³ = (1, 0, 1, 1, 0) with an optimal value of −14 is obtained. Even though the best principal dual is updated to (x ³ , λ ³ ), the best feasible solution is left as an empty set because x ³ is not feasible.

The Lagrange relaxation undergrading in this solution is two. And its next Lagrange multiplier is calculated as follows:
λ ^{4 =} 5 + 0.5 * 2 = 6

The following Lagrange relaxation problems are solved by the quantum anila:
An optimal solution x ⁴ = (1, 0, 0, 1, 0) with an optimal value of −13 is obtained. Since x ⁴ is feasible, the best feasible solution is also updated along with the best principal dual (x ⁴ , λ ⁴ ).

In this current solution, the Lagrange relaxation undergrading norm is zero and the method ends. The best feasible solution x ⁴ = (1,0,0,1,0) and the best principal duality (x ⁴ , λ ⁴ ) = ((1,0,0,1,0), 6) are reported Ru.

  In the context of the present application, in the solution obtained, if no two co-workers have overlapping shifts, the team consisting of workers 1 and 4 and of all teams will have It will be the most productive team.

  An advantage of the disclosed method is that it improves the processing of the system to solve the BPCPPP Lagrange dual. More precisely, the method of the present disclosure is less error prone than the methods of the prior art, which is a considerable advantage.

  It should be noted that the non-volatile computer readable storage medium is further disclosed. A non-volatile computer-readable storage medium is used to store computer-executable instructions that, when executed, cause a digital computer to perform a method for solving the BPCPPP Lagrange dual, the method comprising the steps of: Obtaining the BPCPPP step; and iteratively until convergence is detected: Provide a set of Lagrange multipliers, provide UBQPP representing the Lagrange relaxation of BPCPPP in these Lagrange multipliers, and quantum UBQPP Providing to the anila, obtaining at least one corresponding solution from the quantum anila, and using the at least one corresponding solution to generate a new set of Lagrange multipliers; and after convergence is detected, the corresponding BPCPPP's Known best of Lagrange duals The step of providing all of the best feasible solutions of known the main dual and BPCPPP.

  While the above description relates to the specific embodiments currently considered by the inventors, in a broad sense the present invention is also construed to include functional equivalents to the elements described. It should be.

Item 1
A method for solving Lagrange duals of a binary polynomially constrained polynomial programming problem, said method comprising
Using the processor:
Obtaining a binary multinomial constrained multinomial programming problem;
Iteratively until convergence is detected,
Provide a set of Lagrange multipliers,
Providing an unconstrained binary quadratic programming problem representing the Lagrange relaxation of the binary polynomially constrained polynomial programming problem in the set of provided Lagrange multipliers,
Providing the unconstrained binary quadratic programming problem to the binary optimizer,
Obtain at least one corresponding solution from the binary optimizer,
Generating a set of new Lagrange multipliers using said at least one corresponding solution;
After convergence, the known best primal dual of the Lagrange dual of the corresponding binary multinomial constrained multinomial programming problem and the known best of the binary multinomial constrained multinomial programming problem Providing all of the viable solutions.

Item 2
The method according to claim 1, wherein acquiring the binary polynomially constrained multinomial programming problem comprises:
Obtaining data representing an objective function that is a polynomial;
Obtaining data representing a polynomial equality constraint;
Obtaining data representing a multiple inequality constraint.

Item 3
The method according to any one of Items 1 to 2, wherein the binary multinomial constrained polynomial programming problem is obtained from at least one of a user, a computer, a software package, and an intelligent agent.

Item 4
In the method according to any one of Items 1 to 3, the step of obtaining the binary polynomial constrained polynomial programming problem further includes initializing software parameters and obtaining a step size subroutine. The way, including.

Item 5
The method according to claim 4, wherein initializing the software parameter is:
Parameterization of the general degree reduction form of the general Lagrange relaxation of the binary polynomially constrained polynomial programming problem for the binary quadratic function parameterized with the Lagrange multiplier using the original variables and auxiliary variables Methods, including providing as a family.

Item 6
The method according to claim 4, wherein initializing the software parameter is:
Providing a general embedding for the general order reduction form of the general Lagrange relaxation of the binary polynomially constrained polynomial programming problem;
Providing an embedded solver function to provide a list of solutions;
Providing one of an initial value and a default value for the Lagrange multiplier,
Providing an error tolerance for the convergence condition;
Providing an integer representing the limit value for the total number of iterations and the limit value for the non-improved iterations.

Item 7
The method according to claim 1, wherein the initial Lagrange relaxation of the binary polynomially constrained polynomial programming problem is generated using a set of initial Lagrange multipliers.

Item 8
The method according to claim 1, wherein the at least one corresponding solution is used to generate a Lagrange dual undergradient of the binary polynomially constrained polynomial programming problem.

Item 9
The method according to claim 1, wherein the at least one corresponding solution is used to generate at least one outer approximation cut for linear reformulation on Lagrange duals of the binary polynomially constrained polynomial programming problem. How to generate.

Item 10
The method according to any one of Items 1 to 9, wherein providing corresponding solutions to Lagrange duals of the binary polynomially constrained polynomial programming problem includes storing the corresponding solutions in a file The way, including.

Item 11
Digital computer:
Central processing unit;
Display device;
A communication port for operatively connecting the digital computer to the binary optimizer;
A memory unit comprising an application for solving Lagrange duals of a binary polynomially constrained polynomial programming problem, said application comprising:
Instructions for obtaining a binary polynomially constrained multinomial programming problem;
Providing a set of Lagrange multipliers iteratively, representing Lagrange relaxations of the binary polynomially constrained multi-term programming problem in these Lagrange multipliers, and providing an unconstrained binary quadratic programming problem, the communication port Providing the unconstrained quadratic programming problem to the binary optimizer using: obtaining at least one corresponding solution from the binary optimizer via the communication port, the at least one corresponding solution An instruction to use to generate a new set of Lagrange multipliers until convergence is detected;
After convergence is detected, the known best principal dual of Lagrange duals of the corresponding binary multinomial constrained multinomial programming problem and the known of the binary multinomial constrained multinomial programming problem known A memory unit including instructions for providing all of the best possible solutions;
A digital computer, comprising: a data bus for interconnecting the central processing unit, the display unit, the communication port, and the memory unit.

Item 12
A non-volatile computer readable storage medium for storing computer executable instructions that, when executed, causes a digital computer to perform a method for solving Lagrange duals of a binary polynomially constrained polynomial programming problem. And the method is:
Obtaining a binary multinomial constrained multinomial programming problem;
Iteratively until convergence is detected:
Provide a set of Lagrange multipliers,
Provide an unconstrained binary quadratic programming problem representing the Lagrange relaxation of the binary polynomially constrained polynomial programming problem in these Lagrange multipliers,
Provide the above unconstrained quadratic programming problem to the binary optimizer,
Obtain at least one corresponding solution from the binary optimizer,
Generating a new set of Lagrange multipliers using the at least one corresponding solution;
After convergence is detected, the known best principal dual of Lagrange duals of the corresponding binary multinomial constrained multinomial programming problem and the known of the binary multinomial constrained multinomial programming problem known Providing all of the best possible solutions.

Item 13
A method for solving Lagrange duals of a binary polynomially constrained polynomial programming problem, said method comprising:
Obtaining a binary multinomial constrained multinomial programming problem;
Iteratively until convergence is detected,
Provide a set of Lagrange multipliers,
Providing an unconstrained binary quadratic programming problem representing the Lagrange relaxation of the binary polynomially constrained polynomial programming problem in the set of provided Lagrange multipliers,
Providing the unconstrained binary quadratic programming problem to the binary optimizer,
Obtain at least one corresponding solution from the binary optimizer,
Generating a set of new Lagrange multipliers using said at least one corresponding solution;
After convergence, the known best primal dual of the Lagrange dual of the corresponding binary multinomial constrained multinomial programming problem and the known best of the binary multinomial constrained multinomial programming problem Providing all of the feasible solutions.

Item 14
14. The method according to any one of items 1 to 10 and 13, wherein the binary optimizer comprises a quantum aniler.

Item 15
12. A digital computer according to item 11, wherein the binary optimizer comprises a quantum aniler.

Item 16
The non-volatile computer readable storage medium according to Item 12, wherein the binary optimizer includes a quantum aniler.

Claims (16)

A method for solving Lagrange duals of a binary polynomially constrained polynomial programming problem, said method comprising
Using the processor:
Obtaining a binary multinomial constrained multinomial programming problem;
Iteratively until convergence is detected,
Provide a set of Lagrange multipliers,
Providing an unconstrained binary quadratic programming problem representing the Lagrange relaxation of the binary polynomially constrained polynomial programming problem in the set of provided Lagrange multipliers,
Providing the unconstrained binary quadratic programming problem to the binary optimizer,
Obtain at least one corresponding solution from the binary optimizer,
Generating a set of new Lagrange multipliers using said at least one corresponding solution;
After convergence, the known best primal dual of the Lagrange dual of the corresponding binary multinomial constrained multinomial programming problem and the known best of the binary multinomial constrained multinomial programming problem Providing all of the viable solutions. The method according to claim 1, wherein the step of obtaining the binary polynomially constrained polynomial programming problem comprises:
Obtaining data representing an objective function that is a polynomial;
Obtaining data representing a polynomial equality constraint;
Obtaining data representing a multiple inequality constraint.   The method according to any one of the preceding claims, wherein the binary polynomially constrained polynomial programming problem is obtained from at least one of a user, a computer, a software package and an intelligent agent.   4. A method according to any one of the preceding claims, wherein the step of obtaining the binary polynomially constrained polynomial programming problem comprises initializing software parameters and obtaining a step size subroutine. In addition, the method. The method according to claim 4, wherein initializing the software parameters is
Parameterization of the general degree reduction form of the general Lagrange relaxation of the binary polynomially constrained polynomial programming problem for the binary quadratic function parameterized with the Lagrange multiplier using the original variables and auxiliary variables Methods, including providing as a family. The method according to claim 4, wherein initializing the software parameters is
Providing a general embedding for the general order reduction form of the general Lagrange relaxation of the binary polynomially constrained polynomial programming problem;
Providing an embedded solver function to provide a list of solutions;
Providing one of an initial value and a default value for the Lagrange multiplier,
Providing an error tolerance for the convergence condition;
Providing an integer representing the limit value for the total number of iterations and the limit value for the non-improved iterations.   The method of claim 1, wherein the initial Lagrange relaxation of the binary polynomially constrained polynomial programming problem is generated using a set of initial Lagrange multipliers.   The method according to claim 1, wherein the at least one corresponding solution is used to generate Lagrange dual undergradients of the binary polynomially constrained polynomial programming problem.   The method according to claim 1, wherein the at least one corresponding solution is used to at least one outer approximation cutting for linear reformulation on Lagrange duals of the binary polynomially constrained polynomial programming problem. How to generate   10. A method according to any one of the preceding claims, wherein the provision of corresponding solutions to Lagrange duals of the binary polynomially constrained polynomial programming problem comprises storing the corresponding solutions in a file. Method, including. Digital computer:
Central processing unit;
Display device;
A communication port for operatively connecting the digital computer to the binary optimizer;
A memory unit comprising an application for solving Lagrange duals of a binary polynomially constrained polynomial programming problem, said application comprising:
Instructions for obtaining a binary polynomially constrained multinomial programming problem;
Providing a set of Lagrange multipliers iteratively, representing Lagrange relaxations of the binary polynomially constrained multi-term programming problem in these Lagrange multipliers, and providing an unconstrained binary quadratic programming problem, the communication port Providing the unconstrained quadratic programming problem to the binary optimizer using: obtaining at least one corresponding solution from the binary optimizer via the communication port, the at least one corresponding solution An instruction to use to generate a new set of Lagrange multipliers until convergence is detected;
After convergence is detected, the known best principal dual of Lagrange duals of the corresponding binary multinomial constrained multinomial programming problem and the known of the binary multinomial constrained multinomial programming problem known A memory unit including instructions for providing all of the best possible solutions;
A digital computer, comprising: a data bus for interconnecting the central processing unit, the display unit, the communication port, and the memory unit.   The digital computer according to claim 11, wherein the binary optimizer comprises a quantum aniler. A non-volatile computer readable storage medium for storing computer executable instructions that, when executed, causes a digital computer to perform a method for solving Lagrange duals of a binary polynomially constrained polynomial programming problem. And the method is:
Obtaining a binary multinomial constrained multinomial programming problem;
Iteratively until convergence is detected:
Provide a set of Lagrange multipliers,
Provide an unconstrained binary quadratic programming problem representing the Lagrange relaxation of the binary polynomially constrained polynomial programming problem in these Lagrange multipliers,
Provide the above unconstrained quadratic programming problem to the binary optimizer,
Obtain at least one corresponding solution from the binary optimizer,
Generating a new set of Lagrange multipliers using the at least one corresponding solution;
After convergence is detected, the known best principal dual of Lagrange duals of the corresponding binary multinomial constrained multinomial programming problem and the known of the binary multinomial constrained multinomial programming problem known Providing all of the best possible solutions. The non-volatile computer readable storage medium of claim 13 , wherein the binary optimizer comprises a quantum aniler. A method for solving Lagrange duals of a binary polynomially constrained polynomial programming problem, said method comprising:
Obtaining a binary multinomial constrained multinomial programming problem;
Iteratively until convergence is detected,
Provide a set of Lagrange multipliers,
Providing an unconstrained binary quadratic programming problem representing the Lagrange relaxation of the binary polynomially constrained polynomial programming problem in the set of provided Lagrange multipliers,
Providing the unconstrained binary quadratic programming problem to the binary optimizer,
Obtain at least one corresponding solution from the binary optimizer,
Generating a set of new Lagrange multipliers using said at least one corresponding solution;
After convergence, the known best primal dual of the Lagrange dual of the corresponding binary multinomial constrained multinomial programming problem and the known best of the binary multinomial constrained multinomial programming problem Providing all of the feasible solutions. The method according to any one of claims 1-10 and 15 , wherein the binary optimizer comprises a quantum aniler.